Math & Engineering
Arch Calculator
Calculate dimensions and properties of circular arches including span, rise, radius, and area.
Reviewed by the calculator.uk.com Team · Last reviewed 6 May 2026 · Our methodology
Enter the span and rise to calculate arch dimensions
Related to Arch Calculator
The Arch Calculator is a specialized tool that helps determine the key dimensions and properties of a circular arch based on its span and rise. It uses geometric principles and trigonometry to calculate important measurements that are essential for architectural and engineering applications.
Key Measurements
The calculator takes two primary inputs: the span (the horizontal distance between the arch's endpoints) and the rise (the vertical distance from the span to the arch's highest point). From these inputs, it calculates the radius, arc length, area, and central angle of the arch.
Mathematical Formulas
The calculator uses the following formulas: - Radius (R) = (span² + 4×rise²)/(8×rise) - Central Angle (θ) = 2 × arcsin(span/(2×R)) - Arc Length (L) = R × θ - Area (A) = R² × θ/2
Understanding the results of your arch calculations is crucial for practical applications in construction and design. Each calculated value provides important information about different aspects of the arch structure.
Radius
The radius determines the curvature of the arch. A larger radius creates a flatter arch, while a smaller radius results in a more curved arch. This measurement is crucial for ensuring proper structural support and aesthetic appearance.
Arc Length
The arc length represents the actual curved distance along the arch. This measurement is essential for material calculations and construction planning, as it tells you how much material you'll need for the curved portion of the arch.
Area
The calculated area represents the space enclosed by the arch and its span. This is useful for calculating material coverage, decorative elements, or structural load distributions.
1. What is the difference between span and rise in an arch?
The span is the horizontal distance between the two endpoints of the arch, while the rise is the vertical distance from the base of the arch to its highest point. These two measurements are fundamental in defining the shape and size of an arch.
2. How does the radius affect the arch's appearance?
The radius determines how curved or flat the arch appears. A larger radius creates a gentler curve and a flatter arch, while a smaller radius creates a more pronounced curve and a taller arch relative to its span.
3. Why is the central angle important?
The central angle helps determine the portion of a full circle that the arch represents. This is crucial for construction planning and understanding the arch's geometric properties. It affects both the visual appearance and structural behavior of the arch.
4. How can I use the arc length calculation?
The arc length is essential for material estimation and construction planning. It tells you the actual length of material needed to create the curved portion of the arch, which is longer than the straight-line distance between the endpoints (span).
5. What is the scientific source for this calculator?
The radius formula used here, R = (span² + 4×rise²) ÷ (8×rise), comes from the standard geometric relationship between a circle's chord, the height of its segment, and its radius. The central angle and arc length are then found with ordinary circle trigonometry. These are general-purpose formulas taught in geometry and used throughout architecture and construction, not specific to any single proprietary method or publication.